Evaluating Feynman Integrals

By: Vladimir A. Smirnov

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Hardcover | 1 February 2005

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266 Pages

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24.13 x 15.88 x 1.27

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The problem of evaluating Feynman integrals over loop momenta has existed from the early days of perturbative quantum field theory.

Although a great variety of methods for evaluating Feynman integrals has been developed over a span of more than fifty years, this book is a first attempt to summarize them. Evaluating Feynman Integrals characterizes the most powerful methods, in particular those used for recent, quite sophisticated calculations, and then illustrates them with numerous examples, starting from very simple ones and progressing to nontrivial examples.

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"The methods of analytically evaluating Feyman integrals are described in detail. The book is a first attempt to systematically expose variety of methods developed during fifty years ... . It will be useful for students of last years, postgraduate students and theoretical physicists specializing in quantum field theory." (Michael B. Mensky, Zentralblatt MATH, Vol. 1098 (24), 2006)

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Introduction	p. 1
Notation	p. 8
References	p. 9
Feynman Integrals: Basic Definitions and Tools	p. 11
Feynman Rules and Feynman Integrals	p. 11
Divergences	p. 14
Alpha Representation	p. 18
Regularization	p. 20
Properties of Dimensionally Regularized Feynman Integrals	p. 24
References	p. 29
Evaluating by Alpha and Feynman Parameters	p. 31
Simple One- and Two-Loop Formulae	p. 31
Auxiliary Tricks	p. 34
Recursively One-Loop Feynman Integrals	p. 34
Partial Fractions	p. 35
Dealing with Numerators	p. 36
One-Loop Examples	p. 38
Feynman Parameters	p. 41
Two-Loop Examples	p. 43
References	p. 52
Evaluating by MB Representation	p. 55
One-Loop Examples	p. 56
Multiple MB Integrals	p. 63
More One-Loop Examples	p. 65
Two-Loop Massless Examples	p. 71
Two-Loop Massive Examples	p. 81
Three-Loop Examples	p. 92
More Loops	p. 98
MB Representation versus Expansion by Regions	p. 102
Conclusion	p. 105
References	p. 106
IBP and Reduction to Master Integrals	p. 109
One-Loop Examples	p. 109
Two-Loop Examples	p. 114
Reduction of On-Shell Massless Double Boxes	p. 120
Conclusion	p. 127
References	p. 130
Reduction to Master Integrals by Baikov's Method	p. 133
Basic Parametric Representation	p. 133
Constructing Coefficient Functions. Simple Examples	p. 138
General Recipes. Complicated Examples	p. 146
Two-Loop Feynman Integrals for the Heavy Quark Potential	p. 152
Conclusion	p. 162
References	p. 163
Evaluation by Differential Equations	p. 165
One-Loop Examples	p. 165
Two-Loop Example	p. 170
Conclusion	p. 173
References	p. 176
Tables	p. 179
Table of Integrals	p. 179
Some Useful Formulae	p. 185
Some Special Functions	p. 187
References	p. 189
Summation Formulae	p. 191
Some Number Series	p. 192
Power Series of Levels 3 and 4 in Terms of Polylogarithms	p. 197
Inverse Binomial Power Series up to Level 4	p. 198
Power Series of Levels 5 and 6 in Terms of HPL	p. 200
References	p. 204
Table of MB Integrals	p. 207
MB Integrals with Four Gamma Functions	p. 207
MB Integrals with Six Gamma Functions	p. 214
Analysis of Convergence and Sector Decompositions	p. 221
Analysis of Convergence	p. 221
Practical Sector Decompositions	p. 229
References	p. 232
A Brief Review of Some Other Methods	p. 233
Dispersion Integrals	p. 233
Gegenbauer Polynomial x-Space Technique	p. 234
Gluing	p. 235
Star-Triangle Relations	p. 236
IR Rearrangement and R^*	p. 237
Difference Equations	p. 240
Experimental Mathematics and PSLQ	p. 241
References	p. 243
List of Symbols	p. 245
Index	p. 247
Table of Contents provided by Publisher. All Rights Reserved.

Evaluating Feynman Integrals

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Hardcover

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